Intuitionistic logic from a metainferential perspective

ÁLVAREZ LISBOA, MIGUEL AGUSTÍN

Seminario; WIP SEMINAR; 2021

Buenos Aires Logic Group

In a series of recent papers, Barrio, Pailos and Szmuc (from now on, BPS) defended the idea that a logic should not be identified with a set of logical laws, nor with a set of inferences, but with a whole collection of metainferences of all levels. A metainference is ``an inference from inferences to inference(s)'', and a metainference of level n+1 is an inference from metainferences of level n to inferences of level n. Nowadays there is a blooming comunity discussing this understanding of logical identity that plays a key role in a broader project in philosophy of logic that they dubbed `The Buenos Aires Plan' (BA-Plan).This metainferentialist view of logical consequence (from now on, MVLC) stemmed from the debate on substructural solutions to semantic paradoxes. The metainferential logic that BPS endorse is based on ST, a non transitive logic based on the Strong Kleene algebra. This logic was proven interesting in this context by Cobreros et al., who made a strong case for its adequacy in handling the paradoxes of Sorites and the Liar. The key concept in all these solutions is to dissociate the valuation standards for premises and conclusions, and maybe for this reason the philosophical approach has been largely remained model-theoretic: although these logics (both ST and its metainferential counterpart, ST-omega) have proof-theoretical presentations, it is fair to say that the whole understanding of ``the ST phenomenon'' is, for the most part, strongly alethic.Because of this, the logics one usually encounters in this discussion are the ones philosophically concerned with truth. The most common are classical logic (CL) and the logic of the paradox (LP), two well-known inhabitants of the Strong-Kleene realm for which several results of recapture and collapse has been proven. And for the exact same reason, one logic that one never encounters is intuitionistic logic (IL): not only because it is not philosophically motivated by reflections on truth, but also because it is a logic that is not an inhabitant of the Strong-Kleene realm nor any other finite algebra. But, as I see it, the MVLC is broader, and thus somewhat independent, from the particular debate it stemmed. Therefore, the purpose of this paper is to take MVLC and consider IL from its perspective. This will take us to the presentation of a metainferential intuitionistic logic (MIL) is the appropriate one. To be sure, I do not want to make an intuitionistic version of a metainferential logic; I want to present a metainferential version of IL. The exercise shall illuminate us about IL and also teach us some things about the logic ST-omega and some philosophical questions around the BA-Plan.